Based on J.-E. Pin's answer here, I'd like to know the following:
Question. Is there a name for pairs of elements $(a,b)$ of a semigroup $S$ satisfying $\forall x,y \in S : axbayb = axyb$?
Motivation. This is equivalent to saying that the function $S \rightarrow S$ given by $x\mapsto axb$ is a morphism of semigroups.
Examples.
If $S$ has an identity element $1$, then $ba = 1$ implies that $(a,b)$ has the aforementioned property. Thus if $a$ is a unit, then $(a,a^{-1})$ always has this property.
If $s \in S$ is central (i.e. commutes with everything) and satisfies $s^4 = s^2$ then $(s,s)$ has the aforementioned property. In particular, this means that every central idempotent has this property.
Theorem. If $(a,b)$ and $(a',b')$ have this property, then so too does $(a'a,bb').$
Proof. Since $x \mapsto axb$ and $x \mapsto a'xb'$ are morphisms, hence the composite $x \mapsto a'axbb'$ is a morphism. Thus $(a'a,bb')$ has the desired property. This can also be checked directly, though it's a bit hard to follow: $$a'axybb' = a'axbaybb' = a'axbb'a'aybb'$$